In recent years, a number of authors have studied Ramanujan-type supercongruences from the perspective of \(q\) -analogues.
There are various methods to prove \(q\) -congruences, such as, properties of roots of unity [1]},
basic hypergeometric series identities [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, the \(q\) -Wilf-Zeilberger method [8]}, [9]}.
For instance, using the Bailey transformation formula (see [10]}): for \(|\alpha ^2ab/q^2|<1\) ,
\(&\sum _{k=0}^{\infty }\frac{(1+\alpha q^{2k})(\alpha ^2,q^2/a,q^2/b;q^2)_k(-q,\alpha q/\lambda ;q)_k(-\alpha ^2\lambda ab)^kq^{k^2-3k}}{(1+\alpha )(q^2,\alpha ^2a,\alpha ^2b;q^2)_k(\alpha ,-\lambda ;q)_k}\nonumber \\&=\frac{(\alpha ^2q^2,\alpha ^2ab/{q^2};q^2)_\infty }{(\alpha ^2a,\alpha ^2b;q^2)_\infty }\,{}_{4}\phi _{3}\!\left[\begin{array}{c}q^2/a,\ q^2/b,\ \lambda ,\ q\lambda \\q\alpha ,\, q^2\alpha ,\, \lambda ^2\end{array};q^2,\, \frac{\alpha ^2ab}{q^2}\right],\) 
